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14.123 Microeconomic Theory IIISpring 2009

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MIT 14.123 (2009) by Peter EsoLecture 7: Game Theory Basics

1. Rationalizability, Dominance, Equilibrium

2. SPE in “almost-perfect information” games

3. Application to Bargaining

Read: FT Chapters 1, 2.1, 3.1-3.5, 4.1-4.2, 4.4.Solve: FT 2.2, 2.5, 4.9, due March 5, 2009

Starred slides (*) are for Recitation / Review.

Introduction, Prerequisites

• Game theory studies strategic decision making.

• You should already know (from 14.122 or other course):

– Extensive and normal form representations of a game, pure and mixed strategies;

– Concepts of Nash equilibrium (pure or mixed), subgameperfect equilibrium, dominant and dominated strategies;

– Imperfect and incomplete information; normal form of Bayesian games and Bayesian equilibrium;

– Some well-known games, applications, such as Prisoners’ Dilemma, Cournot competition.

• MWG Chapters 7, 8.A-8.E, 9.A-B.

14.123 Lecture 7, Page 2

We Will Learn in Lectures 7-13

7. Refresher (slides marked *) and other fundamental concepts: Rationalizability and Iterated Strict Dominance; Existence and properties of Nash equilibrium; Subgame-perfection in games with “almost-perfect information”; Single-deviation principle.

8. Advanced equilibrium concepts in games with imperfect or incomplete information.

9. Sender-receiver games; costly and cheap-talk signaling.

10. Auction games.

11. Global games.

12. Repeated games & Folk Theorems with perfect observation.

13. Miscellaneous topics, review.


*Normal Form Games

• DEF: A (normal form) game is a triplet (N, S, u):

– N = {1, . . . , n} is the set of players.

– S = S1 × … × Sn is the set of pure strategy profiles, where Si is the set of pure strategies of player i.

– u = (u1,…,un) is the players’ vNM utility functions, where ui : S → R is player i’s vNM utility function.

• DEF: A normal form game is finite if S and N are finite.

• Note that it is implicitly assumed that all players have expected utility preferences, and that the game is common knowledge.


*Mixed Strategies and Beliefs • Denote Δ(X) the set of probability distributions on set X.

• DEF: A mixed strategy of player i is σi ∈ Δ(Si) such that strategy si ∈ Si is chosen with probability σi(si).

• DEF: Mixed strategy profile: σ ∈ Δ(S).

• DEF: Independent profile: σ = σ1 × ... × σn ∈ Δ(S1) × ... × Δ(Sn).

• DEF: Player i’s conjecture (beliefs) about the other players’ strategies is σ-i ∈ Δ(S-i).

Player i may believe that the others’ strategies are correlated!

• Expected payoffs: ui(σ) = Eσ(ui) = Σs∈Sσ(s)ui(s).


Rationality and Dominance

• DEF: Player i is rational if he maximizes his expected payoff given his beliefs.

• DEF: si * is a best reply to a belief σ-i if

∀si ∈ Si : ui(si *,σ-i) ≥ ui(si,σ-i).

• Bi(σ-i) = (mixed) best replies to σ-i.

• DEF: σi strictly dominates si if

∀s-i ∈ S-i : ui(σi,s-i) > ui(si,s-i).

• DEF: σi weakly dominates si if ∀s-i ∈ S-i : ui(σi,s-i) ≥ ui(si,s-i), with a strict inequality for some s-i.


Rationality and Dominance

• THM: In a finite game, si * is never a best reply to any (possibly

correlated) conjecture σ-i if and only if si * is strictly dominated

(by a possibly mixed) strategy.

• That is, si * is a best reply to some σ-i iff it is not strictly dominated.

■ One direction (say, “necessity”):

Suppose si * ∈ Bi(σ-i).

∀si, ui(si *,σ-i) ≥ ui(si,σ-i)

∀σi, ui(si *,σ-i) ≥ ui(σi,σ-i)

No σi strictly dominates si *. ■


Proof, Cont’d (“Sufficiency”)• Lemma (Separating Hyperplanes Theorem): Let C and D be

non-empty, disjoint, convex subsets of m such that C is closed. Then, ∃r ∈ m\{0} : ∀x∈cl(D), ∀y∈C, r⋅x ≥ r⋅y.

■Proof in MWG, Math Appendix, p. 948. ■

• Let S-i = {s-i 1,…, s-i

m}, and denote ui(si,.) = (ui(si,s-i 1),…, ui(si,s-i

m)), U = {ui(si,.) | si ∈ Si}, and co(U) = {ui(σi,.) | σi ∈ Δ(Si)}.

• Define D = {x∈m | xk > ui(si *,s-i

k), ∀k=1,…,m}.

• If si * is not strictly dominated, then co(U) and D are disjoint.

• By the Sep. Hyperplanes Thm, ∃r: ∀σi, ui(si *,σ-i) ≥ ui(σi,σ-i),

where σ-i(s-ik) = rk/(r1+…+ rm). ■


Rationalizability

• Let S0 = S, and for all k=1,2,…, let Ski = Bi(Δ(Sk-1

-i)).

• DEF: Player i’s (Correlated) Rationalizable strategies are

S∞ i = ∩k≥0 Sk

i .

• DEF: Independent Rationalizability: Let si ∈ Ski if si ∈ Bi(Πj≠iσj)

where σj ∈ Δ(Sk-1 j) ∀j. σi is I-Rationalizable if σi ∈ Bi(Δ(S∞

-i)).

• THM: Correlated Rationalizability is equivalent to the Iterated Elimination of Strictly Dominated Strategies in finite games.

■ Follows from the THM on slide #7. ■

• If the game and rationality are “common knowledge”, then each player plays a rationalizable strategy.


Application: Cournot Duopoly • The Game:

– Simultaneously, each firm i∈{1,2} produces qi units at a constant marginal cost c (no fixed cost).

– Both firms sell their production at price P = max{0,1-q1-q2}.

• Best response:

– If i believes j produces qj, then Bi(qj) = (1-c-qj)/2.

• Therefore, – If i knows that qj ≤ q, then qi ≥ (1-c-q)/2.

– If i knows that qj ≥ q, then qi ≤ (1-c-q)/2.


Application: Cournot Duopoly • Rationalizability / Iterated strict dominance:

– We know that qj ≥ q0 = 0.

– Then, qi ≤ q1 = (1-c-q0)/2 = (1-c)/2 for each i;

– Then, qi ≥ q2 = (1-c-q1)/2 = (1-c)(1-1/2)/2 for each i;

– …

– Then, qn ≤ qi ≤ qn+1 or qn+1 ≤ qi ≤ qn where

qn+1 = (1-c-qn)/2 = (1-c)(1-1/2+1/4-…+(-1/2)n)/2.

– As n→∞, qn → (1-c)/3.

• THM: Cournot Duopoly (as defined in our example) has a unique rationalizable outcome.


Cournot Best Responses

q2

1-c

B1(q2)

(1-c)/2

(1-c)/4 B2(q1)

0 1-c 1-c 1-c q14 2


Elimination, Round 1

q2

1-c

(1-c)/2

(1-c)/4

0 1-c 1-c 1-c q14 2

B2(q1)

B1(q2)


Elimination, Round 2

q2

1-c

(1-c)/2

(1-c)/4

0 1-c 1-c 1-c q14 2

B2(q1)

B1(q2)


More Rounds of Elimination

q2

1-c

(1-c)/2

(1-c)/4

0 1-c 1-c 1-c q14 2

B2(q1)

B1(q2)


*Nash Equilibrium

• DEF: σ* = (σ*1,…,σ*

n) is a Nash Equilibrium if ∀i, σ* i ∈ Bi(σ*

-i), where Bi are player i’s (potentially mixed) best replies.

• Alternatively: ∀i, ∀si ∈ Si : ui(σ* i,σ*

-i) ≥ ui(si,σ* -i).

• THM (Existence): Each finite game has a (possibly mixed) Nash equilibrium σ*.

• Follows from another existence theorem:

• THM: Let each Si be a convex, compact subset of a Euclidean space and each ui continuous in s and quasi-concave in si.

Then, there exists a Nash equilibrium s ∈ S.


Continuity and Fixed Points

• DEF: A correspondence F : X → 2Y, where X is compact and Y bounded, is upper-hemicontinuous if F has a closed graph:

[xm → x & ym → y & ym ∈ F(xm)] ⇒ y ∈ F(x).

• THM (Berge’s Maximum Theorem): Assume f : X × Z → Y is continuous and X, Y, Z are compact. Let F(x) = arg maxz∈Zf(x,z). Then F is non-empty, compact-valued, and upper-hemicontinuous.

• THM (Kakutani’s Fixed-Point Theorem): Let X be a convex, compact subset of m and let F : X → 2X be a non-empty, convex-valued correspondence with a closed graph. Then there exists x ∈ X such that x ∈ F(x).

■ See MWG Math Appendix. ■


Proof of the Existence Theorem

■ Let F : S → 2S be the “best reply” correspondence:

Fi(s) = Bi(s-i).

– By Berge’s Maximum Theorem: F is non-empty and has a closed graph.

– By quasi-concavity, F is convex valued.

– By Kakutani’s Fixed-Point Theorem: F has a fixed point: s* ∈ F(s*).

– s* is a Nash equilibrium. ■


Upper-Hemicontinuity of NE

• X, S are compact metric spaces, ux(s) continuous in x∈X and s∈S.

• Let NE(x) be the set of Nash equilibria of (N,S,ux).

• Let PNE(x) be the set of pure Nash equilibria of (N,S,ux).

• THM: NE and PNE are upper-hemicontinuous.

■ Note that Δ(Si) is compact and ux(σ) is continuous in (x,σ).

Suppose xm→x, σm∈NE(xm), but σm → σ ∉ NE(x). (NE not uhc)

Then, ∃i, si: ux(si,σ-i) > ux(σ).But then uxm(si,σ-i

m) > uxm(σm) for large m, so σm ∉ NE(xm), á. ■

• Corollary: If S is finite, then NE is non-empty, compact-valued, and upper-hemicontinuous.


*Extensive Form

N = Players A tree Payoffs

Information partition (of non-terminal nodes)

Player map

Nature’s moves Initial node

Terminal Nodes(1,0)

(0,1) (0,0)

(2,0)

(1,1)

(0,2)

(1,0)

(0,1)

1

1

2

2

0 .5

.5


*Definitions, Notation

• An extensive form game is described by a set of players N, a tree (X,<), utility functions (on terminal nodes) un for n ∈ N, an information partition (of the non-terminal nodes) H, a player map i, and probability distributions for Nature’s moves.

• DEF: A tree is a directed graph (X,<) such that � there exists an initial node , i.e. < x for all x ≠ ;

� < is transitive: (x < y, y < z) ⇒ x < z; � < is asymmetric: x < y ⇒ not (y < x); � < satisfies arborescence: (x < z, y < z) ⇒ (x < y or y < x).

• Denote Z = { z | ∄ x with z < x} the set of terminal nodes and ui : Z→ the players’ utility functions (terminal payoffs).


*Definitions, Notation

• Denote A(x) the actions available at x ∈ X\Z. • DEF: An information partition is a partition H of X\Z such that

x,y ∈ h ∈ H ⇒ A(x) = A(y) =: A(h). • DEF: A player map is a function i : H → N {0}.

Interpretation: i(h) moves at h ∈ H. Let hi ∈ Hi = {h | i(h) = i }.

• If i(h) = 0, then Nature’s moves are determined by a probability distribution on A(h).

• DEF: A player has perfect recall if he does not “forget” what he “knew” or “did” at prior information sets.

The formal definition in terms of the structure of the tree and the information partition is in FT, p. 81.


*Normal & Reduced Normal Forms

• Define the strategy set for player i as Si = ∏hi∈Hi A(hi). A strategy is a complete, contingent plan.

• The outcome of s ∈ S is O(s) ∈ Δ(Z). (Mixtures over terminal nodes because of Nature’s moves.)

• Define i’s payoff from s as ui(s) = ui(O(s)).

• DEF: The normal form of an extensive form game is (N, S, u).

• Strategies si and si ’ are equivalent if ∀s-i , O(si,s-i) = O(si’,s-i).

• Let SiR = {one strategy from each equivalence class}.

• DEF: The reduced normal form is (N, SR, u).


Mixed & Behavior Strategies

• DEF: Player i’s mixed strategy is σi ∈ Δ(Si). Probability of playing si ∈ Si ≡ ∏ A(hi) is denoted by σi(si).

• DEF: A behavior strategy is bi : Hi → Δ(A(hi)) . bi(hi) is a probability distribution over actions at hi .

• THM (Kuhn): In an extensive form game with perfect recall, mixed and behavior strategies are equivalent. More precisely: A unique behavior strategy is equivalent to a class of mixed strategies that generate it.

■ For a proof and more discussion, see FT Chapter 3.4.3. ■


Games with Almost-Perfect Info

• Or: “Multi-stage games with observable actions”, FT Chapter 4.

• Stages k = 0, 1, 2, … . (Can be finite or infinite horizon.)

• At stage k = 0, – players N(∅) are active; – each i ∈ N(∅) simultaneously plays action ai

0 ∈ Ai(∅).

• At k = 1, 2, … and history hk = (a0, a1, …, ak-1), – players N(hk) are active; – history hk = (a0, a1, …, ak-1) is commonly known; – each i ∈ N(hk) simultaneously plays action ai

k ∈ Ai(hk).

• Payoffs u(h) received at each terminal history h = (a0, a1, … ).


Special Cases, Examples

• Games of Perfect Information: At each k and hk, exactly one player is active.

• (Infinitely) Repeated Games with Observable Actions:

� t = 0, 1, 2,…, T ≤ ∞ .

� G = a simultaneous action game. � At each t ≤ T, stage game G is played, and players observe

the actions taken before t.

� Payoffs = Discounted sum (alternatively: average) of payoffs received in the stage game.


Special Cases, Examples

• Alternating offer bargaining (Rubinstein-Ståhl): Split a dollar.

• Risk-neutral players, discount factors δ1 and δ2.

• At “time” t, if t is even, then – Player 1 offers (x1,1-x1), – Player 2 accepts / rejects, – If Player 2 accepts, then game ends with payoffs (x1,1-x1);

otherwise proceed to t + 1.

• At t odd, the roles of players 1 and 2 are reversed.

• Note that each t corresponds to two stages.


More Notation, Definitions

• h = (a0, a1,…, ak-1, ak,…) ⇒ hk = (a0, a1,…, ak-1) (Truncated terminal histories.)

• G(k,h) = subgame starting at h = (a0, a1,…, ak-1).

• sk,h = restriction of strategy s to G(k,h).

• u(s|k,h) = E[ u(O(s)) | k, h ] (= E[u(sk,h)] in G(k,h).)

• u(σ|k,h) = E[ u(σ) | k, h] = E[u(σk,h)].

• DEF: G is continuous at infinity if for all i, ∀ε > 0, ∃k such that

[ hk = ĥk ] ⇒ |ui(h) – ui(ĥ)| < ε.


Single Deviation Principle

• THM: Let G be a game of almost-perfect information; if infinite horizon, then continuous at infinity. A strategy-profile s = (si,s-i) (pure, for simplicity) is SPE if and only if there is no strategy si ’ of player i that agrees with si except at a single k and history h and gives a better response to s-i in G(k,h) than si does.

• Interpretation: If si (played against s-i) cannot be improved with a single-period deviation at k (going back to whatever si prescribes right after the deviation, at k+1), then (si,s-i) is an SPE.

■ Finite horizon: Suppose profile s satisfies single-dev principle, but isn’t subgame perfect: There is stage k, history h, such that some si ’ is a better response to s-i in subgame G(h,k) than si is.


Single Deviation Principle

• Let k’ be the last stage where si and si ’ differ for some history h’; k’ > k, finite. Construct si ” so that it agrees with si ’ before stage k’ and agrees with si from stage k’ on.

• Since si ’ also agrees with si from stage k’+1 on and si cannot be improved (against s-i) in just one stage, si ” is as good a response to s-i as si ’ is from stage k on, for any history including h.

• Successively construct si ’” (agrees with si ’ before k’-1, with si

from k’-1 on), etc. Each improves on si ’; eventually get si , á.

• Infinite horizon with continuity at infinity: Given any payoff difference between si ’ and si (against s-i), one can ignore stages k > K for K large. Back to finite horizon. ■


SPE in Alt. Offer Bargainingx2 1 THM: In the unique SPE,

• P1 offers y* in each turn z*

δ2 • P2 accepts x1 ≤ y1

*

• P2 offers z* in each turn

• P1 accepts x1 ≥ z1*.

y*

δ10 1 x1


Proofx2 1 Use Single-deviation Principle:

1. If P2 rejects an offer at t, z* then she gets z2

* at t+1. δ2 Hence accepting iff x2 ≥

δ2 z* 2 y* δ2 z2

* ≡ y2* is optimal at t.

2. At t, it is optimal for P1 to offer y * = argmaxx1∈[0,1]{x1 | 1–x1≥y*

2}.

0 δ1 y* 1 δ1 1

x1


Iterated Conditional Dominance

• DEF: Action aik(h) is conditionally dominated for history h if

every σi with σi(aik(h)|h) > 0 is strictly dominated by some σi:

∀s-i: ui(σi,s-i|k,h) > ui(σi,s-i|k,h).

• “Iterated Conditional Dominance”: iteratively eliminate all conditionally dominated actions. (Def. 4.2 in FT, pp. 128-129.)

• Finite games of perfect information: ↔ SPE.

• Infinitely repeated games (discount factor near 1): No bite. Any action can be optimal on a temporary basis if it induces ‘good’ actions by opponents and other actions induce ‘bad’ continuations.

• THM: In a game of perfect information (finite or infinite), every SPE profile survives iterated conditional dominance.


Cond. Dominance in Bargaining

0 δ1(1-δ2) δ1 1

Feasible offers (x1,1-x1); discount δi.

1. P1 accepts x1 > δ1 if offered; P2 accepts x1 < 1 – δ2 if offered.

2. P1 only offers x1 ≥ 1 – δ2 ; P1 rejects x1 < δ1(1 – δ2) if offered, since P2 accepts 1 – δ2 next round.

P2 only offers x1 ≤ δ1; P2 rejects x1 > 1 – δ2(1 – δ1).

… x1

x2

y*

1

δ2

δ2(1-δ1)


Iteration:• Suppose that after k rounds of iterated conditional dominance,

P1 accepts any x1 > zk and P2 accepts any x1 < yk, with zk > yk.

• After one more round, P1 only offers x1 ≥ yk and rejects all x1 < δ1yk ; P2 only offers x1 ≤ zk and rejects all x1 > 1 – δ2(1 – zk).

• Claim: After the next round of deletion of dominated actions, Player 1 accepts x1 > zk+1 ≡ δ1(1 – δ2) + δ1δ2 zk.

■ If P1 refuses x1, the best continuation for him is that P2 accepts 1 – δ2(1 – zk) next period, yielding δ1[1 – δ2(1 – zk)] = zk+1. ■

• Similarly, P2 accepts x1 < yk+1 ≡ (1 – δ2) + δ1δ2 yk.

• z∞ = δ1(1-δ2)/(1-δ1δ2); y∞ = (1-δ2)/(1-δ1δ2).

• P2 accepts x1 < y∞, rejects x1 > 1 – δ2(1 – z∞) = y∞; unique SPE.


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